A circular drum 1.8 m diameter and 1.2 m height is submerged with its axis vertical and its upper end at a depth of 1.8 m below water level. Determine total pressure on top, bottom and curved surfaces of the drum, resultant pressure on the whole surface, and depth of centre of pressure on the curved surface.

Submerged Circular Drum Problem

Problem Statement

A circular drum 1.8 m diameter and 1.2 m height is submerged with its axis vertical and its upper end at a depth of 1.8 m below water level. Determine: (i) total pressure on top, bottom and curved surfaces of the drum, (ii) resultant pressure on the whole surface, and (iii) depth of centre of pressure on the curved surface.

Given Data

Drum Diameter, $ D = 1.8 \, \text{m} $
Drum Height, $ H = 1.2 \, \text{m} $
Depth of top surface, $ h_{top} = 1.8 \, \text{m} $
Fluid is water, $ \rho = 1000 \, \text{kg/m}^3 $

Solution

(i) Total Pressure on Individual Surfaces

First, calculate the area of the top and bottom circular surfaces.

$$ A = \frac{\pi}{4} D^2 $$ $$ A = \frac{\pi}{4} (1.8)^2 $$ $$ A \approx 2.5447 \, \text{m}^2 $$

Total Pressure on Top Surface ($F_{top}$):

$$ F_{top} = \rho g A h_{top} $$ $$ F_{top} = 1000 \times 9.81 \times 2.5447 \times 1.8 $$ $$ F_{top} \approx 44933 \, \text{N} $$

Total Pressure on Bottom Surface ($F_{bottom}$):

$$ \text{Depth of bottom surface, } h_{bottom} = h_{top} + H $$ $$ h_{bottom} = 1.8 + 1.2 = 3.0 \, \text{m} $$

$$ F_{bottom} = \rho g A h_{bottom} $$ $$ F_{bottom} = 1000 \times 9.81 \times 2.5447 \times 3.0 $$ $$ F_{bottom} \approx 74888 \, \text{N} $$

Total Pressure on Curved Surface:

For a vertically submerged object with a symmetrical curved surface, the horizontal hydrostatic forces acting on it are in equilibrium. Therefore, the resultant horizontal force on the curved surface is zero.

$$ F_{curved} = 0 \, \text{N} $$

(ii) Resultant Pressure on the Whole Surface

The resultant pressure (or net force) on the entire drum is the vector sum of all forces. Since the horizontal forces are zero, the resultant force is the net vertical force.

$$ F_{resultant} = F_{bottom} - F_{top} \text{ (acting upwards)} $$ $$ F_{resultant} = 74888 - 44933 $$ $$ F_{resultant} = 29955 \, \text{N} $$

This is the buoyant force, which can be verified by calculating the weight of the displaced water.

$$ V_{drum} = A \times H = 2.5447 \times 1.2 = 3.0536 \, \text{m}^3 $$ $$ F_{buoyant} = \rho g V_{drum} = 1000 \times 9.81 \times 3.0536 $$ $$ F_{buoyant} \approx 29956 \, \text{N} $$

The results match, confirming the resultant force is the buoyant force.

(iii) Depth of Centre of Pressure on Curved Surface

The pressure on the curved surface varies linearly from top to bottom. The centre of pressure is the centroid of the trapezoidal pressure distribution acting on the projected vertical area.

$$ \text{Pressure at top of drum, } p_{top} = \rho g h_{top} = 9.81 \times 1.8 = 17.658 \, \text{kPa} $$ $$ \text{Pressure at bottom of drum, } p_{bottom} = \rho g h_{bottom} = 9.81 \times 3.0 = 29.43 \, \text{kPa} $$

The depth of the centre of pressure ($h^*$) is the centroid of this pressure distribution.

$$ h^* = h_{top} + \frac{H}{3} \left( \frac{p_{top} + 2 p_{bottom}}{p_{top} + p_{bottom}} \right) $$ $$ h^* = 1.8 + \frac{1.2}{3} \left( \frac{17.658 + 2 \times 29.43}{17.658 + 29.43} \right) $$ $$ h^* = 1.8 + 0.4 \left( \frac{76.518}{47.088} \right) $$ $$ h^* = 1.8 + 0.4 \times 1.625 $$ $$ h^* = 1.8 + 0.65 $$ $$ h^* = 2.45 \, \text{m} $$

Final Results:

(i) Pressure on Top: $ \approx 44.93 \, \text{kN} $; on Bottom: $ \approx 74.89 \, \text{kN} $; on Curved Surface: 0 kN.

(ii) Resultant Pressure on Drum: $ \approx 29.96 \, \text{kN} $ (upwards).

(iii) Depth of Centre of Pressure on Curved Surface: $ h^* = 2.45 \, \text{m} $.

Explanation of Concepts

Pressure on Horizontal Surfaces: The force on the top and bottom circular surfaces is found by multiplying the uniform pressure at that depth ($P = \rho g h$) by the area of the circle. The force on the bottom is greater because it is deeper.

Pressure on Curved Surfaces: For a closed, symmetrical body like a vertical cylinder, the horizontal pressure forces acting on the curved sides are equal and opposite at every depth, so they cancel each other out completely. The net horizontal force is zero.

Resultant Force (Buoyancy): The net force on a fully submerged object is the buoyant force. It is the difference between the upward force on its bottom surface and the downward force on its top surface. This is equivalent to the weight of the fluid displaced by the object's volume.

Centre of Pressure on Curved Surface: Although the net force on the curved surface is zero, a pressure distribution still exists. The "centre of pressure" represents the average depth where this pressure acts. It is found by calculating the centroid of the trapezoidal pressure diagram that acts on the side of the drum.